THE ISOSCELES TRIANGLE

Book I. Proposition 5

EVEN THOUGH we practice the proofs of the theorems, they become hollow exercises unless we see that they are true. What we see becomes the proof -- there should be no gap between seeing and proving. Upon reading the enunciation, the setting out, and the specification of a theorem, the student should see that it is true. (Difficult to see might be the Pythagorean theorem, and perhaps that is why so many proofs have been offered.)

We are now ready to prove the well-known theorem about isosceles triangles, namely that the angles at the base are equal. And we can see that. Compare the isosceles triangle on the left

with the scalene triangle on the right.

There are several ways to prove this theorem, and we shall give the clever proof by Pappus, a Greek mathematician who followed Euclid in Alexandria. When we learn how to bisect an angle, we will see another proof. But this amusing proof is based clearly on what we see.